The Operator Product Expansion

Nolan Smyth 6/10/2020

QFT 3 - Final Presentation Outline

1. The Big Picture

2. Renormalization Group Refresher/Setup

3. Introducing the Operator Product Expansion

4. Callen-Symanzik Equation for Wilson Coeﬃcients

5. An example: QCD corrections to weak decays

1 The Big Picture The Big Picture

We’ve spent much of this quarter learning to deal with divergences, typically those of an ultraviolet nature. These divergences led to shifting in mass parameters and coupling constants through the process of renormalization, but were otherwise rather innocuous. Renormalization gave us a great deal of freedom in that we were free to deﬁne our theory at any arbitrary momentum scale. But in order to describe the the correct physical theory, the renormalized Green’s functions must be identical to the bare Green functions, up to the ﬁeld strength renormalization.

(n) −n/2 (n) GR (µ; x1, ..., xn) = Z Gb (x1, ..., xn)

2 The Big Picture

We then asked, how would our Green’s functions change if we looked at a different renormalization scale? This led to the Callen-Symanzik (renormalization group) equations. Solving these, we calculated how coupling constants and fields evolve with momentum scale. Until now, we’ve utilized correlation functions to perform this analysis. But we will see that it is particularly useful to study the renormalization flow at the level of the operators themselves. But to do so, we will need to deal with the non-locality of operator products. This is where we will meet the Operator Product Expansion (OPE).

3 The Big Picture

After motivating the OPE, we will see that all of the interesting behavior at high energy (short distances) depends only on Wilson coeﬃcents. These can be calculated once and then used for any process since they do not depend on any external ﬁelds that appear in a Green’s function. We will then use the OPE to look at QCD renormalization of the weak interactions. Since QCD has asymptotic freedom, we will see that the OPE allows us to determine the renormalization corrections in terms of the anomalous dimensions of the operators.

4 Renormalization Group Refresher/Setup RG

The bare and renormalized Green’s functions are related by

(n) −n/2 (n) GR (µ; x1, ..., xn) = Z Gb (x1, ..., xn) (1)

When varying the renormalization scale µ, we must also change Z and the coupling constants (here we’ll use λ) in order for these Green’s functions to represent the same physical theory. This can be written generally as

dG (n) ∂G (n) ∂G (n) ∂λ R = R + R (2) dµ ∂µ ∂λ ∂µ

5 RG

By taking the derivative of the left hand side of (1) with respect to µ, and using the fact that the bare Green’s function does not depend on the renormalization scale, we ﬁnd

dG (n) n ∂Z R = − G (n). (3) dµ 2Z ∂µ R

Combining these results, we end up with the renormalization group equation

h ∂ ∂ i µ + β + nγ G (n) = 0, (4) ∂µ ∂λ R ∂λ µ ∂Z where β ≡ µ ∂µ and γ ≡ 2Z ∂µ control how the coupling constant and ﬁeld φ vary with µ respectively.

6 RG

Correlation functions containing composite operators (operators evaluated

at the same spacetime point) must also be renormalized. Ob = ZOOR . Thus, we may write the relationship between the bare and renormalized Green’s functions as

(n;1) −n/2 −1 (n;1) GR (µ; x1, ..., xn; y) = Z ZO Gb (x1, ..., xn; y), (5)

where

(n;1) GR (µ; x1, ..., xn; y) = hφ(x1)...φ(xn)O(y)i. (6)

7 RG

By the same treatment as before, requiring the bare correlation function to be independent of the renormalization scale µ, we end up with the following equation

h ∂ ∂ i µ + β + nγ + γ G (n;1) = 0, (7) ∂µ ∂λ O R where the anomalous dimension of the composite operator is deﬁned as

µ ∂ZO γO ≡ (8) ZO ∂µ

8 RG

It would be nice to study the renormalization ﬂow at the level of the operators. But the behavior of composite operators is unwieldy

(A1(x1)A2(x2)...An(xn) is singular when the arguments coincide (x1 = x2... = xn)). For example, consider the humble free-ﬁeld propagator

Z d 4p i h0|T φ(x)φ(y)|i = D (x − y) = e−ip·(x−y). (9) F (2π)4 p2 − m2 + i

In the limit of x → y (forming a composite operator), this is divergent! To make our lives easier, we use the OPE to stick all the diﬃcult behavior inside the Wilson coeﬃcients.

9 Introducing the Operator Product Expansion OPE

Let’s say we’re looking at a process that involves two operators, O1 and O2, which act on points separated by a small distance x. Let’s also imagine that there are external physical states located much further away

φ(yi ). The amplitude for this process can be calculated from the Green’s function (where I’m writing all products of operators as time ordered)

G12(x; , y1...ym) = hO1(x)O2(0)φ(y1)...φ(ym)i. (10)

10 OPE

In a correlation function like hO1(x)O2(0)φ(y1)...φ(ym)i, since the local operators φ(yi ) are far away from 0, x, in the limit of x → 0, the operator product looks like a single local operator. It can, in fact, produce the most general local disturbance. Thus, we can expand it in terms of the basis which contains all possible local operators at 0.

11 OPE

The operator product expansion (OPE) states that the product of local operators evaluated at diﬀerent points, in the limit that those points approach each other, can be written as a sum over composite (local) operators:

X lim O1(x)O2(0) = Cn(x)On(0). (11) x→0 n

This is useful because it holds at the level of operators; the OPE is independent of external states. In other words, regardless of any ﬁelds that might appear in a Green’s function, once you have calculated a OPE once, you may then use it to calculate matrix elements for any process.

12 Callen-Symanzik Equation for Wilson Coeﬃcients Wilson Coeﬃcients

Using the OPE, our Green’s function can be expanded as

X n G12(x; y1, ...ym) = C12(x)Gn(y1, ...ym), (12) n where

Gn(y1, ...ym) = hOn(0)φ(y1)...φ(ym)i. (13)

Notice that, indeed, all of the dependence on x is in the Wilson coeﬃcients (the C(x)’s). But these, in turn, only depend on the operators themselves since

X lim O1(x)O2(0) = Cn(x)On(0). (14) x→0 n

13 Wilson Coeﬃcients

We expect the Wilson coeﬃcients to depend on the renormalization scale. So to determine this dependence, we write the RG equation for the renormalized Green’s function

h ∂ ∂ i µ + β + nγ + γ + γ G (n) = 0. (15) ∂µ ∂λ 1 2 12

The RG equation for the Green functions Gn that show up on the right side of (12) give us

h ∂ ∂ i µ + β + nγ + γ G = 0, (16) ∂µ ∂λ n n

where γn is the anomalous dimension of the operator On.

14 Wilson Coeﬃcients

In order to satisfy

X n G12(x; y1, ...ym) = C12(x)Gn(y1, ...ym), n the Wilson coeﬃcients must obey the equation

h ∂ ∂ i µ + β + γ + γ − γ C (n) = 0. (17) ∂µ ∂λ 1 2 n 12

This shows that our previous assumption of the dependence of the coefficients on µ to be valid. Most importantly, this shows that the Wilson coefficients only depend on the anomalous dimensions of the operators, but not on the specific correlation function! Thus the number of fields φ and their anomalous dimension does not come into play.

15 x-dependence of Wilson Coeﬃcients

Let’s say the dimensions of the operators O1, O2, and On are d1, d2, and dn respectively. Then we may write the Wilson coeﬃcients as

(n) 1 ñ C12 (x; µ) = C12(µ|x|), (18) |x|d1+d2−dn ñ where C12 is dimensionless. To find the structure of the coefficient, we introduce the running coupling λ(1/|x|), defined by dλ(p, λ) = β(λ), λ(µ, λ) = λ d ln(p/µ) which lets us write the solution as

(n) (n) C12 (λ(1/|x|)) C12 (x; µ) = |x|d1+d2−dn (19) Z µ 0 h dp 0 0 0 i × exp 0 [γn(λ(p )) − γO1 (λ(p )) − γO2 (λ(p ))] 1/|x| p

(n) 16 where C12 is a perturbative function of the running coupling. x-dependence of Wilson Coeﬃcients

(n) (n) C12 (λ(1/|x|)) C12 (x; µ) = |x|d1+d2−dn Z µ 0 h dp 0 0 0 i × exp 0 [γn(λ(p )) − γO1 (λ(p )) − γO2 (λ(p ))] 1/|x| p

We see that the short distance behavior is singular if dn < d1 + d2. On the other hand, if dn > d1 + d2, the contribution goes to zero so we do not need to consider operators of this kind in the OPE. In class, we saw that QCD was asymptotically free. That is, the coupling goes to zero at short distances. Let’s examine the Wilson coeﬃcients further in this case.

17 QCD Wilson Coeﬃcients

To ﬁrst order, any anomalous dimension can be written as g 2 γO ≡ −aO (4π)2 for some numerical constant aO. Thus we have

α γ − γ − γ = (a + a − a ) s . (20) d 1 2 1 2 n (4π)

At one loop, the running coupling αs is simply

4π α (Q2) = (21) s Q2 β0 ln( 2 ) ΛQCD

where β0 is the coeﬃcient of the ﬁrst term in the Taylor expansion of the QCD β function and Q is the momentum scale of the given process.

18 QCD Wilson Coeﬃcients

This leads us to

(n) 2 2 an−a1−a2 C g(1/|x|) hln(1/|x| Λ )i 2β C (n)(x; µ) = 12 QCD 0 . (22) 12 d1+d2−dn 2 2 |x| ln(µ /ΛQCD)

In particular, note that when dn = d1 + d2, the main source of the |x| dependence is the powers of the logarithmic terms.

19 Operator Mixing

It is possible for quantum corrections to mix operators under the evolution of the renormalization scale. In that case, we must consider the more general form of the Callen-Symanzik equation for the Wilson Coeﬃcients

h ∂ ∂ i X µ + β + γ + γ C j − γ C i = 0, (23) ∂µ ∂λ 1 2 12 ij 12 i

where γij is now a matrix.

20 An example: QCD corrections to weak decays Overview - QCD corrections to weak decays

We will see that the matrix element for a QCD process mediated by an operator O(x) will receive corrections which depend on the momentum scale of the process and the renormalization scale. These corrections will depend on the anomalous dimension of the operator which takes the form

g 2 γ = −a . (24) O O (4π)2 After solving the Callen-Symanzik equation, we will ﬁnd the QCD renormaliztion factor

2 2 ln(µ /Λ ) aO /2b0 , (25) ln(Q2/Λ2) where Q is the momentum scale of the process mediated by O, µ is the

renormalization scale used to deﬁne the operator normalization, and b0 is 2nf the coeﬃcient of the QCD β function b0 = 11 − 3 .

21 QCD corrections to weak decays

At low energy, the weak interaction between quarks and leptons in the standard model can be described by Fermi’s 4-fermion theory. The interaction term in the Lagrangian is

4GF µ ν† Lint ≈ √ J (0)J (0) + h.c., (26) 2 L L µ where JL is the left handed charged current. In the full electroweak theory, the product of currents would be non-local and the Lagrangian µ ν† would contain instead JL (0)Dµν (0, x)JL (x). Using the OPE we will study how the non-local electroweak operators get replaced by local ones.

22 The Trivial Cases - QCD corrections to weak decays

Since leptons do not couple directly to gluons at one loop, there will be no QCD corrections to purely leptonic weak interactions. For a semi-leptonic weak interaction (involving both a leptonic current and a quark current), again the leptons don’t receive corrections to ﬁrst order. Moreover, the quark current has an anomalous dimension of zero so it is not aﬀected by strong interactions. The only non-trivial case is non-leptonic weak interactions, i.e. weak interactions between quark currents. Let’s look at these now.

23 Strange Decay - QCD corrections to weak decays

Let’s look at the weak decay of the strange quark. In the standard model, the interactions between charged currents containing leptonic and quark terms has the form

g 2 L = JµD (0, x)Jν†(x) + h.c., (27) int 2 L µν L

where Dµν is the propagator of the W boson. At low momenta, we can make the approximation

1 −1 2 2 → 2 , (28) k − mW mW in which case, the weak interaction becomes an eﬀective local vertex (Fermi theory). We will ﬁnd the consequences of the original composite interaction by working with this local operator.

24 Strange Decay - QCD corrections to weak decays

µ In Fermi theory, we have the coupling (d Lγ uL)(uLγµsL). So let’s then look at the OPE of the product of the currents

µ µ µ A1 (x) ≡ d Lγ uL, A2 (0) = uLγµsL. (29) These will be expanded on the operators

µ µ O1 ≡ (d Lγ uL)(uLγµsL), O2 ≡ (d Lγ sL)(uLγµuL). (30)

Note that d1 + d2 = dn, so the x dependence of the Wilson coeﬃcients comes entirely from the logarithms at one loop, as we saw previously.

25 Strange Decay - QCD corrections to weak decays

(n) To determine the Wilson coeﬃcients C12 for the operators On, we ﬁrst need to calculate the anomalous dimensions of all the involved operators.

Since A1 and A2 are conserved, their anomalous dimension is zero

γA1 = γA2 = 0! This is a consequence of the Ward identity and can be veriﬁed easily. As we saw in class, a conserved currents imply that the corresponding charges (Q = R d 3xj 0) form a Lie algebra

c [Qa, Qb] = fabQC , (31) which by dimensional analysis tells us that conserved currents never acquire an anomalous dimension.

26 Strange Decay - QCD corrections to weak decays

To get the anomalous dimension of the operators O1, O2, we need to 2 calculate all the order g corrections. For O1, these come from the diagrams

27 Strange Decay - QCD corrections to weak decays

The calculation of the previous diagrams is not particularly important. See Gelis 7.3 for the details. The takeaway is that the operators mix, causing the anomalous dimensions for the operators to form a non-diagonal matrix

! g 2 −2 6 γ = (32) ij (4π)2 6 −2

Thus we must solve a coupled Callen-Symanzik equation h ∂ ∂ i X µ + β C j − γ C i = 0, (33) ∂µ ∂λ 12 ij 12 i by rotating to a diagonal basis. This is satisﬁed for operators

1 1 O ≡ [O − O ], O = [O + O ]. (34) 1/2 2 1 2 3/2 2 1 2

28 Strange Decay - QCD corrections to weak decays

The eigenvalues of the matrix γij are

g 2 g 2 γ = −8 , γ = 4 . (35) 1/2 (4π)2 3/2 (4π)2

To first order then, we find the values for the Wilson coefficients at a −1 distance scale of x ≈ mw

2 2 4 hln(m /Λ )i β C 1/2(m−1; µ) = w QCD 0 12 w ln(µ2/Λ2 ) QCD (36) ln(m2 /Λ2 ) − 2 3/2 −1 h w QCD i β0 C12 (mw ; µ) = 2 2 . ln(µ /ΛQCD) The superscripts on these operators refer to their isospin quantum

numbers. In turns out that O1/2 can mediate processes that change the isospin by 1/2 but not by 3/2.

29 Kaon Decays - QCD corrections to weak decays

Now that we ﬁnally have the QCD corrections to strange quark decays, let’s quickly see how this applies to the K meson (the lightest hadron with an s quark). The process K 0 → π+π− changes the isospin by 1/2. The process K + → π+π0 changes isospin by 3/2. Experimentally, the former occurs at a much higher rate.

For ΛQCD ≈ 150 MeV, µ = mK , we ﬁnd the operator O1/2 receives an enhancement whereas the operator O3/2 receives a suppression. Thus the weak decay of the s quark favors isospin 1/2 processes by a factor of ≈ 3. Experimentally, the relative diﬀerence bewtween the processes is ≈ 20. Getting the exact rates requires non-perturbative QCD, but the power of the OPE is that we were able to make this qualitative prediction using only perturbative methods.

30 Referencesi

References

[1] Fran¸coisGelis. Quantum ﬁeld theory: from basics to modern topics. Cambridge University Press, Cambridge, United Kingdom ; New York, NY, 2019. ISBN 978-1-108-48090-1. [2] Michael Edward Peskin and Daniel V. Schroeder. An introduction to quantum ﬁeld theory: student economy edition. Westview Press, Boulder, Colorado, 2016. ISBN 978-0-8133-5019-6. OCLC: 927402141. [3] Antonio Pineda. Breakdown of the operator product expansion of deep inelastic scattering in the ’t Hooft model. page 11.

31 References ii

[4] Matthew Dean Schwartz. Quantum field theory and the standard model. Cambridge University Press, New York, 2014. ISBN 978-1-107-03473-0. [5] Kenneth G. Wilson and Wolfhart Zimmermann. Operator product expansions and composite field operators in the general framework of quantum field theory. Communications in Mathematical Physics, 24 (2):87–106, 1972. ISSN 0010-3616, 1432-0916. URL https://projecteuclid.org/euclid.cmp/1103857739. Publisher: Springer-Verlag.